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We present methods for constructing any target coupling graph using limited global controls in an Ising-like quantum spin system. Our approach is motivated by implementing the quantum approximate optimization algorithm (QAOA) on trapped ion quantum hardware to find approximate solutions to Max-Cut. We present a mathematical description of the problem and provide approximately optimal algorithmic constructions that generate arbitrary unweighted coupling graphs with $n$ nodes in $O(n)$ global entangling operations and weighted graphs with $m$ edges in $O(m)$ operations. These upper bounds are not tight in general, and we formulate a mixed-integer program to solve the graph coupling problem to optimality. We perform numeric experiments on small graphs with $n\le8$ and show that optimal sequences, which use fewer operations, can be found using mixed-integer programs. Noisy simulations of Max-Cut QAOA show that our implementation is less susceptible to noise than the standard gate-based compilation.